Subgroups — #27403097

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	2	yes	maximal
2	#20275491	C₂	1	6	yes	maximal
3	#20274853	C₂	1	3	yes	maximal
4	C₄×#13133153	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	6	no	—
7	Q₈×#88322	—	2	2	no	—
8	C₂×#13133153	C₂²	1	6	yes	—
9	#16810647	C₂²	1	6	yes	—
10	C₄×#11719917	C₂²	1	6	yes	—
11	#16810388	C₂²	1	6	yes	—
12	#16810388	C₂²	1	3	yes	—
13	#16810388	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₃	C₄	1	6	yes	—
16	C₈×Q₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
19	C₂×#11719917	C₂³	1	6	yes	—
20	C₂×#11719917	C₂³	1	2	yes	—
21	#13133153	—	2	12	no	—
22	C₄⋊#88322	—	2	6	no	—
23	C₄⋊#88322	—	2	6	no	—
24	#13133128	—	2	6	no	—
25	C₄×#88322	—	2	6	no	—
26	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	D₄	1	3	yes	—
32	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	Q₈	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
36	C₄⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	C₄⋊C₄	1	3	yes	—
41	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	6	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
50	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
51	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
54	C₄²×C₅₃	C₂×D₄	1	3	yes	—
55	C₄²×C₅₃	C₂×D₄	1	3	yes	—
56	#11719917	—	2	2	no	—
57	#11719917	—	2	6	no	—
58	#11719917	—	2	6	no	—
59	#11719917	#41	1	4	yes	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	6	no	—
62	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
63	Q₈×C₅₃	—	2	6	no	—
64	Q₈×C₅₃	#108	1	4	yes	complemented
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	4	3	no	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
74	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
75	C₈×C₅₃	—	2	6	no	—
76	C₈×C₅₃	—	2	2	no	—
77	#88322	—	2	4	no	—
78	#88322	—	4	6	no	—
79	#22746	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₅₃	#557	1	2	yes	—
84	C₄×C₅₃	—	2	3	no	—
85	C₄×C₅₃	—	2	3	no	—
86	C₄×C₅₃	—	2	3	no	—
87	C₄×C₅₃	C₂×#108	1	6	yes	—
88	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
89	C₄×C₅₃	#519	1	1	yes	characteristic
90	C₄×C₅₃	#548	1	2	yes	—
91	Q₈×M₁₆	—	53	2	no	—
92	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
93	#1950	—	53	6	no	—
94	#1837	—	53	3	no	—
95	C₄×#108	—	53	3	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	#2351	1	1	yes	characteristic
98	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
99	C₂×C₅₃	#2916	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
101	#446	—	106	6	no	—
102	C₈×Q₈	—	106	2	no	—
103	C₈×Q₈	Dic₅₃	1	6	yes	—
104	C₈×Q₈	Dic₅₃	1	2	yes	—
105	C₂×#108	—	53	6	no	—
106	#423	—	53	6	no	—
107	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
108	C₄×M₁₆	—	53	6	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
110	#341	—	53	3	no	—
111	#341	—	53	6	no	—
112	#341	—	53	1	no	—
113	C₅₃	#22746	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
115	C₂×M₁₆	—	53	2	no	—
116	C₂×M₁₆	—	53	6	no	—
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
119	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
120	C₄×Q₈	D₂₁₂	1	3	yes	—
121	C₄×Q₈	C₂×Dic₅₃	1	3	yes	—
122	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	Dic₁₀₆	1	3	yes	—
124	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
125	#108	—	106	12	no	—
126	C₄⋊C₈	—	106	6	no	—
127	C₄⋊C₈	—	106	6	no	—
128	C₄⋊C₈	C₂×Dic₅₃	1	6	yes	—
129	C₄⋊C₈	C₂×Dic₅₃	1	6	yes	—
130	#97	—	106	6	no	—
131	C₄×C₈	—	106	6	no	—
132	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
133	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₄⋊Dic₅₃	1	3	yes	—
136	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
138	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
139	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
140	M₁₆	—	53	4	no	—
141	M₁₆	—	106	2	no	—
142	M₁₆	—	106	6	no	—
143	M₁₆	—	106	6	no	—
144	C₂×C₈	—	2	6	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	6	no	—
148	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
149	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
150	C₄⋊C₄	C₄⋊Dic₅₃	1	6	yes	—
151	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
152	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
153	C₄²	C₂×D₂₁₂	1	3	yes	—
154	C₄²	C₂×D₂₁₂	1	3	yes	—
155	C₄²	C₂²×Dic₅₃	1	3	yes	—
156	C₄²	C₂²×Dic₅₃	1	3	yes	—
157	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
158	Q₈	—	2	6	no	—
159	Q₈	#13133153	1	4	yes	complemented
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
166	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
168	C₂×C₄	#41×D₅₃	1	2	yes	—
169	C₂×C₄	#13133265	1	2	yes	—
170	C₈	—	2	6	no	—
171	C₈	—	2	2	no	—
172	C₈	—	106	4	no	—
173	C₈	—	212	6	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂²	#16811349	1	2	yes	—
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#13133153	1	6	yes	—
182	C₄	#41×Dic₅₃	1	1	yes	characteristic
183	C₄	#16811101	1	1	yes	characteristic
184	C₄	#16811285	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#20277636	1	1	yes	characteristic
187	C₂	C₂²×#13133153	1	1	yes	characteristic
188	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#22746
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
64. Q₈×C₅₃	#108	semidirect
113. C₅₃	#22746	semidirect
159. Q₈	#13133153	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×#11719917	2
#20275491	2
#20274853	2
C₄×#13133153	2
C₂×C₈×Q₈×C₅₃	2
#22746	53